Categorical Abstract Logic: Hidden Multi-Sorted Logics as Multi-Term Institutions

Babenyshev and Martins proved that two hidden multi-sorted deductive systems are deductively equivalent if and only if there exists an isomorphism between their corresponding lattices of theories that commutes with substitutions. We show that the -institutions corresponding to the hidden multi-sorted deductive systems studied by Babenyshev and Martins satisfy the multi-term condition of Gil-Férez. This provides a proof of the result of Babenyshev and Martins by appealing to the general result of Gil-Férez pertaining to arbitrary multi-term -institutions. The approach places hidden multi-sorted deductive systems in a more general framework and bypasses the laborious reuse of well-known proof techniques from traditional abstract algebraic logic by using “off the shelf” tools

Categorical Abstract Algebraic Logic: Referential π-Institutions

Wójcicki introduced in the late 1970s the concept of a referential semantics for propositional logics. Referential semantics incorporate features of the Kripke possible world semantics for modal logics into the realm of algebraic and matrix semantics of arbitrary sentential logics. A well-known theorem of Wójcicki asserts that a logic has a referential semantics if and only if it is selfextensional. Referential semantics was subsequently studied in detail by Malinowski and the concept of selfextensionality has played, more recently, an important role in the field of abstract algebraic logic in connection with the operator approach to algebraizability. We introduce and review some of the basic definitions and results pertaining to the referential semantics of π-institutions, abstracting corresponding results from the realm of propositional logics

Categorical Abstract Algebraic Logic: Equivalence of Closure Systems

In their famous ``Memoirs monograph, Blok and Pigozzi defined algebraizable deductive systems as those whose consequence relation is equivalent to the algebraic consequence relation associated with a quasivariety of universal algebras. In characterizing this property, they showed that it is equivalent with the existence of an isomorphism between the lattices of theories of the two consequence relations that commutes with inverse substitutions. Thus emerged the prototypical and paradigmatic result relating an equivalence between two consequence relations established by means of syntactic translations and the isomorphism between corresponding lattices of theories. This result was subsequently generalized in various directions. Blok and Pigozzi themselves extended it to cover equivalences between $k$-deductive systems. Rebagliato and Verd\\u27{u} and, later, also Pynko and Raftery, considered equivalences between consequence relations on associative sequents. The author showed that it holds for equivalences between two term $\pi$-institutions. Blok and J\\u27{o}nsson considered equivalences between structural closure operations on regular $M$-sets. Gil-F\\u27{e}rez lifted the author\u27s results to the case of multi-term $\pi$-institutions. Finally, Galatos and Tsinakis considered the case of equivalences between closure operators on ${\bf A}$-modules and provided an exact characterization of those that are induced by syntactic translations. In this paper, we contribute to this line of research by further abstracting the results of Galatos and Tsinakis to the case of consequence systems on ${\bf Sign}$-module systems, which are set-valued functors \SEN:{\bf Sign}\ra{\bf Set} on complete residuated categories ${\bf Sign}$

Federated description logics for the semantic web

The thesis deals with a family of federated description logics for creating modular ontologies in the semantic web. All these logics share modularity, the possibility to reuse concept names and role names by importing, and context-sensitive interpretation of all logical connectives. Apart from the main basic language F-ALCI, we present a lattice-based extension LF-ALCI, a probabilistic extension PF-ALCI and an extension that employs knowledge operators F-ALCIK. All languages are based on the ordinary well-known description logic ALCI

Categorical Abstract Algebraic Logic: Equivalential π-Institutions

The theory of equivalential deductive systems, as introduced by Prucnal and Wroński and further developed by Czelakowski, is abstracted to cover the case of logical systems formalized as π-Institutions. More precisely, the notion of an N-equivalence system for a given π-Institutions is introduced. A characterization theorem for N-equivalence systems, previously proven for N-parameterized equivalence systems, is revisited and a “transfer theorem” for N-equivalence systems is proven. For a π-Institutions I having an N-equivalence system, the maximum such system is singled out and, then, an analog of Herrmann’s Test, characterizing those N-protoalgebraic π-Institutions having an N-equivalence system, is formulated. Finally, some of the rudiments of matrix theory are revisited in the context of π-Institutions, as they relate to the existence of N-equivalence systems

Categorical abstract algebraic logic

In (1) * the theory of algebraizable deductive systems was developed. A deductive system S over a language L is said to be algebraizable if there exists a quasi-variety K, over the same language L, and translations from the sentences of the system into equations and vice-versa that, roughly speaking, simulate the deduction over the system in the equational deduction over the quasi-variety and vice-versa and are inverses of each other. In (3) this notion of algebraizability was shown to be a special case of the, so-called, equivalence of deductive systems. It is, in fact, the equivalence of S with the very special 2-deductive system (2) associated with the quasi-variety K;One of the main limitations of this framework is the way in which it handles logics with varying signatures, like equational or first-order logic. Before they can be algebraized in this framework, they must be transformed in a rather artificial way to propositional-like structural counterparts;In this thesis, following (4), the [pi]-institution framework (5, 6, 7) is used to extend the theory of algebraizability to logics with varying signatures. The notion of equivalence for [pi]-institutions is introduced. A characterization of equivalence for a special class of [pi]-institutions, the, so-called, term [pi]-institutions, is obtained along the lines of (2), by exploiting the relation between their categories of theories. The notion of an algebraic institution is then defined. Algebraic institutions roughly correspond to equational 2-deductive systems, but the formalism here is categorical rather than universal algebraic so that it can handle the added generality appropriately. These special institutions are used for the algebraization of arbitrary [pi]-institutions. The example of the equational institution illustrates the general theory. Limiting the scope to a subclass of term [pi]-institutions, the, so-called, theory institutions, and requiring the syntax to remain invariant during the algebraization process, an intrinsic characterization is obtained for algebraizability in the spirit of (1), via the introduction of a generalized Leibniz operator. The study of the invariance of some metalogical properties under equivalence of [pi]-institutions and the detailed development of the algebraic theory used in the algebraization of the equational institution conclude the thesis. ftn*Please refer to dissertation for references

Categorical Abstract Algebraic Logic: Equivalential π-Institutions

The theory of equivalential deductive systems, as introduced by Prucnal and Wroński and further developed by Czelakowski, is abstracted to cover the case of logical systems formalized as π-Institutions. More precisely, the notion of an N-equivalence system for a given π-Institutions is introduced. A characterization theorem for N-equivalence systems, previously proven for N-parameterized equivalence systems, is revisited and a “transfer theorem” for N-equivalence systems is proven. For a π-Institutions I having an N-equivalence system, the maximum such system is singled out and, then, an analog of Herrmann’s Test, characterizing those N-protoalgebraic π-Institutions having an N-equivalence system, is formulated. Finally, some of the rudiments of matrix theory are revisited in the context of π-Institutions, as they relate to the existence of N-equivalence systems

Secrecy Logic: Protoalgebraic S-Secrecy Logics

In recent work the notion of a secrecy logic S over a given deductive system S was introduced. Secrecy logics capture the essential features of structures that are used in performing secrecy-preserving reasoning in practical applications. More precisely, they model knowledge bases that consist of information, part of which is considered known to the user and part of which is to remain secret from the user. S-secrecy structures serve as the models of secrecy logics. Several of the universal algebraic and model theoretic properties of the class of S-secrecy structures of a given S-secrecy logic have already been studied. In this paper, our goal is to show how techniques from the theory of abstract alge-braic logic may be used to analyze the structure of a secrecy logic and draw conclusions about its algebraic character. In particular, the notion of a protoalgebraic S-secrecy logic is introduced and several characterizing properties are provided. The relationship between protoalgebraic S-secrecy logics and the protoalgebraicity of their underlying deductive systems is also investigated

Categorical Abstract Algebraic Logic: Pseudo-Referential Matrix System Semantics

This work adapts techniques and results first developed by Malinowski and by Marek in the context of referential semantics of sentential logics to the context of logics formalized as π-institutions. More precisely, the notion of a pseudoreferential matrix system is introduced and it is shown how this construct generalizes that of a referential matrix system. It is then shown that every π–institution has a pseudo-referential matrix system semantics. This contrasts with referential matrix system semantics which is only available for self-extensional π-institutions by a previous result of the author obtained as an extension of a classical result of Wójcicki. Finally, it is shown that it is possible to replace an arbitrary pseudoreferential matrix system semantics by a discrete pseudo-referential matrix system semantics

Categorical Abstract Algebraic Logic: Closure Operators on Classes of PoFunctors

Following work of Palasinska and Pigozzi on partially ordered varieties and quasi-varieties of universal algebras, the author recently introduced partially ordered systems (posystems) and partially ordered functors (pofunctors) to cover the case of the algebraic systems arising in categorical abstract algebraic logic. Analogs of the ordered homomorphism theorems of universal algebra were shown to hold in the context of pofunctors. In the present work, operators on classes of pofunctors are introduced and it is shown that classes of pofunctors are closed under the HSP and the SPPU operators, forming analogs of the well-known variety and quasi-variety operators, respectively, of universal algebra